where \(G_x,G_y\)  are the stabilizer groups of \(x,y\)  and \(T_x\subset T\)  is the subgroup fixing \(x\) . Now according to [1]}, a quasi-separated algebraic stack locally of finite type with affine stabilizers, such that the above exact sequence is split for all \(x\in \mathcal {X}\) , has a smooth \(\mathbb {C}^\ast \) -equivariant presentation from an affine scheme with \(\mathbb {C}^\ast \) -action. Under those assumptions, \(X^0\)  and \(X^\pm \)  are then algebraic spaces (resp. stacks). The functors \((p^\pm _X)_!(\eta ^\pm _X)^\ast \)  are called hyperbolic localization functors. From [2]} there is a natural morphism build from morphisms of the six-functor formalism in the derived category of constructible sheaves, or the derived category of monodromic mixed Hodge modules:
\(S_X:\mathbb {D}(p^-_X)_!(\eta ^-_X)^\ast \mathbb {D}\rightarrow (p^+_X)_!(\eta ^+_X)^\ast \) 
